The cubo-cubic transformation of the smooth quadric fourfold is very special

TL;DR

This paper classifies special self-birational transformations of smooth quadric threefolds and fourfolds, identifying unique examples in each dimension based on specific algebraic geometric configurations.

Contribution

It provides a complete classification of such transformations, revealing only one example exists for each of the quadric threefold and fourfold.

Findings

Unique self-birational transformation for Q^3 via quadrics through a rational quartic

Unique self-birational transformation for Q^4 via cubics through a K3 surface with specific properties

Classification results in a precise geometric description of the transformations

Abstract

We classify special self-birational transformations of the smooth quadric threefold and fourfold, $Q^3$ and $Q^4$. It turns out that there is only one such example in each dimension. In the case of $Q^3$, it is given by the linear system of quadrics passing through a rational normal quartic curve. In the case of $Q^4$, it is given by the linear system of cubics passing through a non-minimal K3 surface of degree $10$ with two skew $(-1)$-lines.